Application of Bézout's Theorem \ Induction

Question

Let $a$ and $b$ be relatively prime integers, and suppose that $c$ is an integer and let $n$ be a positive integer such that $a$ divides the product $b^n c$, then $a$ divides $c$

Im not sure how to approach this problem. If and b are relatively prime i know we they can be expressed as $$1=ar+bs, r,s \in Z$$ And if $a$ divides the product of $b^n c$, that can be expressed as

$$b^n c =ak, k \in Z$$

Im not sure how to approach this problem with induction or how the two expressions work together.

Answer 1

Hint: If $a$ and $b$ are coprime, then also $a$ and $b^n$ are coprime (prove it).

Since $1=ah+b^nk$, for some $h$ and $k$, you can write $$ c=c1=ach+b^nck $$

Answer 2

Hint:

Prove that if $a$ and $b$ are coprime, then, for any $n\ge 0$, $a$ and $b^n$ are coprime too.

Next, apply Gauß' lemma.

Answer 3

Raise the equation $ar+bs=1$ to the power of $n$, collect the first $n$ terms (on the LHS) together and we have \begin{eqnarray*} aR+b^nS=1. \end{eqnarray*} Now multiply this by $c$ and use $b^nc=ak$ \begin{eqnarray*} a(cR+kS)=c, \end{eqnarray*} so $a$ divides $c$.